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MAP

Mathematics Achievement Program

SFSU – High School Collaborative

TOPIC 3: Real Numbers

Real numbers can be represented by points on a number line. Each real number

corresponds to a point on the number line, and each point on the line corresponds to a real

number. Some important subsets of real numbers are: integers, which are the numbers

{

}

−

−

K

K

in the set

,

, 2

1

0 ,

, 1 ,

, 2

; fractions (also called rational numbers), which are

a

numbers of the form

, where a and b are integers and b ? 0; and irrational numbers,

b

which are numbers that cannot be written as fractions.

A square root of a real number a, written

a , is a real number n whose square is

n =

2

a; that is,

a = n when

a

.

=

2

Example 1:

There are two square roots of 16, 4 and – 4, because

4

16

and

( )

−

=

16 = , and

2

. Four is the positive square root of 16, written

4

4

16

−

=

−

– 4 is the negative square root of 16, written

16

4

.

Recall that the square of any real number is either zero or positive. Therefore

only nonnegative real numbers have square roots; negative numbers do not have real-

−

number square roots. For example,

16

is not a real number because there is no real

number whose square is –16.

A perfect square is a number in the set {0, 1, 4, 9, 16, 25, …}. We can use

perfect squares to approximate square roots.

Example 2:

Approximate

17 without using a calculator.

Because 17 is between the perfect squares 16 and 25,

17 is between

16 and

25 . Therefore,

17 is between 4 and 5.

________________________________________________________________________

1. List the perfect squares less than 150.

For exercises 2-5, use perfect squares to approximate the square root without using a

calculator.

2.

7

3.

31

4.

85

5. 128

________________________________________________________________________

The absolute value of a real number a, written a , is the distance from 0 to a on

the number line. The absolute value of a real number is never negative (Why?).

________________________________________________________________________

For exercises 6-11, find the absolute value.

−

−

6.

5

7. 12

8.

12

− −

−

−

9.

5

10. 2 5

11. 5 2

8

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Parent category: Education